Some notes on the notes: when I present a hand-drawn figure in lectures,
please don’t expect that to be reproduced in the notes – I’ll expect you to
sketch those and take some skeletal notes of your own in class. Where I include
graphics, they are usually taken from the web, and if I shown an image on the
screen in class, I generally will put that image in here, since the main reason
I’ll show images is that they are too tough to draw on the board.1 The notes
will be to help guide you, and should not be taken as a transcript of what was
said in class. They are not a replacement for reading the Prialnik book.
Week 1: Motivations for studying stars, and
fundamental measurements of stars
The key goals for this course will be to help you to understand the physics
that governs the structure and evolution of stars, to place stars into a broader
context in astronomy, and to help you to be aware of some of the modern
problems in stellar astrophysics. The emphasis will be on understanding the
physics of stellar structure and evolution.
Some key reasons why stars are important:
• Production of the chemical elements – the Big Bang produces most of
the net hydrogen, helium and lithium in the Universe. By “net” I mean
that, for example, hydrogen fuses to helium in stars, but the helium often
fuses later into heavier elements, and the helium content of the Universe.
Everything else is produced in stars (or in supernovae, which are the last
phase of the lives of massive stars)
• Production of most of the light in the Universe – if you want to understand
where the light comes from you must understand stars. Accretion disks
around supermassive black holes produce a non-trivial minority of the
light in the universe, but stars definitely dominate the total.
• Tracers of the Universe – much of cosmology involves finding tracers of
the distance scale of the Universe. These tracers are almost always stars
of some kind, or stellar explosions of some kind, although there are some
exceptions.
• Dead stars – black holes and neutron stars – are sites for studying extreme
physics. Black holes are one of the few ways in the Universe to understand
strong field general relativity. Neutron stars give important probes of general relativity, of the physics of dense matter (relevant for understanding
particle physics), and of the physics of extremely large magnetic fields.
• Feedback of energy into the interstellar medium – stellar winds and supernovae can help heat and move gas around in galaxies and affect the
evolution of galaxies.
1 I will be negligent in providing citations for these images in some cases. These notes are
not formal academic writing, so don’t take a lack of citation here as a license not to cite figures
yourself. I really should be citing these, too.
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• Nuclear physics – nuclear reactions fuel stars. Thus we can sometimes
hope to learn about nuclear physics based on what happens in stars. A
classic example is the “Hoyle state” of carbon-12. Ordinarily, it should
be hard to produce carbon. However, there’s a resonance for an excited
state of carbon-12 – an alpha particle plus a beryllium nucleus have almost
exactly the same energy as this excited state, making the cross-section for
the reaction much larger than for the reaction that produces the groundstate of carbon-12. Fred Hoyle predicted this on the basis of the fact that
carbon exists in the universe, so there must be some way to crank up the
cross-section for making it. It was only later verified experimentally.
• Must understand star formation to understand planet formation. Also,
though, it’s important to understand stars in detail to find planets around
the stars. The first extrasolar planets were met with worries, because the
radial velocity variations could have been stellar pulsations instead.
• Most importantly, we astronomers, as pure scientists, should be interested
in anything that happens in the sky, and should want to try to be able to
explain it.
Fundamental measurements of stars
Key properties of stars:
Mass, radius, luminosity, temperature, chemical composition (sometimes
consider all elements separately, sometimes lump all metals into metallicity,
sometimes consider the α elements and the iron-peak elements as separate
groups of metals), rotation rate, magnetic field, age, multiplicity
These quantities pretty much define a star. There can be some exceptions
for the cases of stars which have undergone mass transfer. Also, in the very
late stages of evolution of stars, there can be unusual chemical composition
gradients due to convection and pulsations that can cause mixing to take place
in ways that are far different from what happens on the main sequence. But,
for the most part, if you know the values for all the numbers listed above, you
know pretty much everything about a star. Now let’s look at how we go about
measuring these things.
I’ll focus on the model-independent ways to get at these quantities. These
are the ones used to build up a picture of stellar phenomenology (e.g. the
Hertzpsrung-Russell diagram, and an understanding e.g. of what mass stars
have at what place on it). Once a picture of how stars behave has been built up
from the data, model dependent techniques can be developed. These can benefit
from indirect means for measuring quantities which can be easier to make than
the direct measurements, but they cannot be used from the beginning to build
up a picture.
Mass measurements
Mass measurements come from Kepler’s laws in binaries. We covered this
quite extensively in Intro to Astronomy, and you should have also seen Kepler’s
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Laws in classical mechanics as well. We’ll refresh our memory about applying
Kepler’s laws on the problem sheet.
Radius measurements
Eclipsing binaries are a good place to build up measurements of radii of stars.
The durations of the ingresses of the eclipses and of the eclipses themselves tell
you about the radii of the stars. You also need to know what the orbital solutions
are for the stars in the binary, too, to convert a time into a distance travelled.
For certain other types of stars, there are other direct radius measurement
technqiues. For the very nearest and largest stars, one can use optical or infrared
interferometry to image the stars directly.
For pulsating stars, there is a technique called the Baade-Wesselink method.
We didn’t talk about it in lecture, but it’s interesting. It’s also one of these
ideas that is quite straightforward to understand, but which must have taken
a great deal of cleverness to have been thought of in the first place. The basic
idea is as follows: (1) the stellar radius is changing (2) repeated measurements
of the Doppler shift of the star will give the velocity of the envelope of the star
(3) one can subtract off the mean velocity over the pulsation cycle, and then one
gets the velocity of expansion or contraction (4) one can integrate the velocity
of expansion between the time of minimum radius of the star and maximum
radius of the star, giving the difference between the minimum radius and the
maximum radius (5) by looking at the flux divided by σT 4 at minimum radius
versus maximum radius, one can get the ratio of the minimum to the maximum
radius (6) with the difference between the minimum and maximum radius and
the ratio of the radii, one can solve for both radii. This thus turns into a
technique which can give a model independent radius for the stars. It also,
then, provides a calibration for the use of pulsating stars as distance indicators.
Luminosities of stars
Luminosity is straightforward to measure, if you know the distance to a star.
L = 4πd2 F , where L is the luminosity, d is the distance, and F is the flux of the
star. The flux must be corrected for extinction from the interstellar medium.
One must also be careful to get the bolometric corrections right, if one doesn’t
measure the flux across a bandpass which includes all of the star’s light.
Rotation rates
There are a few ways to get rotation rates of stars. One must worry, of
course, about whether the star is rotating differentially. Generally, what will be
easiest to measure is the rotation rate of the surface of the star near the equator.
We know that the Sun has a different rotation rate near the poles than near the
equator (although not by a large factor). We also have good reason to suspect,
from helioseismology, that the core of the Sun rotates at a different rate than
the surface does.
Star spots are a good tracer of a star’s rotation rate, although not all stars
have spots. If a star does have spots, they will darken the star by enough that,
with good data, the brightness of the star will vary periodically on the rotation
period. If you want the rotational velocity of the star, rather than the rotational
period, you still need to get the star’s radius.
Stars with radiative, rather than convective envelopes, and old stars, even
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with convective envelopes, tend to have few or no starspots. Also, the Sun has
a magnetic cycle over which the number of starspots changes dramatically, and
other stars probably do as well, but only about 25 stars other than the Sun have
had their magnetic cycles measured. For stars without spots, other techniques
must be used.
The most widely applicable is to measure the rotational broadening of the
star’s absorption lines, caused by the Doppler effect. However, this gives only
the radial component of the rotational velocity. The inclination angle, which
must be measured to solve for the actual rotational velocity, is almost impossible
to measure. One can thus get the rotational velocities in a statistical sense for
a group of stars, but getting the individual stars rotational velocities is rarely
possible using this method.
Ages of stars
Ages of stars are extremely important to try to measure, but quite hard to
measure.
In star clusters, one can assume the stars are all the same age, and use the
main sequence turn off to get at the age of the star cluster. This is the most
commonly used way to get at the ages of stars, but it’s highly model dependent.
It relies on assumptions about the evolutionary timescales of stars which come
from stellar evolution models. It also relies on the assumption that the stars
really are all the same age. One of the most exciting developments in stellar
physics recently is the discovery that the stars in a cluster are not all the same
age – the discovery of multiple main sequences in certain globular clusters – see
figure .
Neither of these assumptions is too bad – the multiple main sequences are
exciting because they tell us that something in the standard textbooks for the
past century is wrong – but the ages of the different sequences are all within a
few percent. Stellar physics is now very well calibrated, so that’s also not a big
concern. The real problem with using star clusters is that most stars are not
in clusters – or at least not in clusters large enough to measure a turnoff age to
high precision (e.g. in a cluster of 50 stars, there might be quite a large range of
ages possible between the turnoff age for the most massive main sequence star
and that for the least massive evolved star).
As a result, a new technique is needed. About 5 years ago, the concept of
gyrochronology was developed – using stellar rotation rates to measure the ages
of stars. It was well known since the 1960s that the rotation periods of stars
of a given mass scale as the square root of the stellar age. This result comes
from calibrating using clusters. What wasn’t well developed was how to set the
constant of proportionality, nor how precise the ages from the method would
be, but these details have now been recently worked out.
The general thought for why this technique works is that stars slow down
their rotation due to a process called magnetic braking. The stellar wind is
blown through the stellar magnetic field, meaning that the mangetic field is
sweeping through a sea of charged particles. It is not well understood why this
leads to the square root law, but empirically, it does.
Magnetic fields of stars
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Magnetic fields of stars are measured using Zeeman splitting, and some sophisticated variations on Zeeman splitting. Very few measurements exist. We
won’t discuss stellar magnetic fields much in this course, but I thought you
should just be aware of the word so you know what to look up if you ever want
to read about stellar magnetic field variations.
Building up a picture
So, after one knows a bit about stellar distances, one can make a diagram
of luminosity versus some measure of temperature (e.g. colors) of stars. This
diagram is the Hertzsprung-Russell diagram. It can be used to group stars
into temperature classes and luminosity classes, and this was realized early on
- the spectra of stars at different places in the diagram look different, while the
spectra of stars near one another look very similar.
Getting rough measurements of the temperature from the colors is straightforward. The luminosity must then obviously be related to the radius. So a
good question is how the radius affects the spectrum.
It turns out that for a given color, the mass range is quite small compared
to the radius range. The radius is thus related closely to the surface gravity and
hence the surface pressure of the star.
Radius of the star II
Changing the pressure in a stellar atmosphere can have two broad classes
of effects. One is that at high pressure, atoms and molecules are more likely
to re-combine. This is basic thermodynamics – when the density is large, the
Boltzmann factor becomes more important, and the extra entropy associated
with having free particles becomes less important. Small radius, high pressure
cool stars thus have more molecules. This can be seen strikingly in figure .
The other important issue is pressure broadening of lines. In most classes
of stars, the Stark effect can be quite important. The wavelengths expected for
atomic transitions are normally computed in a vacuum, and measured at low
density. At high pressure, there will typically be some free electrons and some
ions near the atoms undergoing atomic transitions. The electrons in the atom
will feel forces not just from their “host” nucleus, but from the free electrons
and ions. This will lead to a perturbation in what the energy levels are in the
atom, broadening the line (since the perturbations will be by different amounts
and with different signs for different atoms).
Pressure broadening is usually the most significant source of broadening in
stars, but its strength varies depending on the specific atomic transition. Other
sources of broadening include thermal doppler broadening, rotational doppler
broadening, and turbulent doppler broadening. The Heisenberg uncertainty
principle also contributes to the width of lines.
The effects of pressure broadening can be seen by comparing the Balmer
lines of a white dwarf with those from a solar-like star in figure . The white
dwarf lines are much broader, reflecting the much higher surface gravity of the
white dwarf.
Temperatures of stars
First, we have to consider how to define the temperature of a star, since
different parts of the star will have different temperatures. Theorists prefer using
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a measure called the effective temperature, defined from Lbol = 4πR∗2 σTef
f.
This has the advantage of being more or less what comes out of theoretical stellar
interior models, but has the disadvantage of requiring knowledge of both the
star’s radius and its bolometric luminosity, both of which are hard to measure.
In the first year course, we talked about how to get temperatures by assuming a star is a blackbody, and measuring the peak wavelength, then using
Wien’s displacement law, λmax T = 2.9mm K. This temperature is called the
color temperature. The main problems with this are that stars aren’t perfect
blackbodies, and that many stars are reddened, and the reddening corrections
aren’t that easy to make just from the colors of the star, so color temperature
measurements are more subject to inaccuracies than temperature measurements
based on spectral lines – but they are a lot easier to make, since spectra are a
lot harder to get than colors.
The most robust observational method for getting temperatures is by looking at the strengths of certain absorption lines. These can be converted into
temperature measurements by using the Saha equation. Meghnad Saha was the
first modern astrophysicist of real significance who came from outside Europe
or North America, so this result has some cultural interest as well.
The Saha equation is:
gi+1
2
∆E
ni+1 ne
(1)
=
exp
ni
Λ
gi
kB T
where n is the density of a species, g is the number of quantum states for a
species, e stands for electrons, i + 1 stands for ions with charge i + 1, i stands
for ions with charge i (or atoms for i = 0), ∆E is the change in energy between
the free and bound state, kB is Boltzmann’s constant, T is the temperature,
and Λ is the De Broglie wavelength of the electrons.
Saha derived this equation from equations in physical chemistry used to
understand equilibrium concentrations of chemicals. There are a few implicit
assumptions. One is that the stellar atmosphere is in local thermodynamic
equilibrium. Another is that the density is small enough that charge screening
by free electrons is not important. These are usually, but not always, reasonably
well satisfied in stellar atmospheres. One can also see that one needs to get an
idea of the pressure before trying to measure the temperature, so that one knows
how to scale the number densities.
Chemical compositions
These will generally be the last things to measure. The reason is that you
might be able to measure the number of atoms in a particular quantum state
from the strength of an absorption line, but to figure out the fraction of the
atoms of that element in that quantum state, you need to know the temperature
and the pressure.
In star clusters, one can look at the color of the main sequence stars or the red
giants stars, or of the integrated light and make a model dependent measurement
of the “chemical composition” of the cluster, under the assumption that the
stars all have the same composition. Like ages, this is a good, but imperfect
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assumption when examined in detail. Additionally, one cannot, from colors
alone, make a good estimate of the relative abundances of different metals.
What we really measure
It is useful to think about what we measure when we look at a star’s spectrum. Basically, what we measure is the blackbody curve’s flux for the temperature of the layer of the star from which the optical depth is unity. When there
is a low opacity at a wavelength, then we see deeper into the star, where the
star is hotter, and we see more flux. A consequence of this is that the relation
between the equivalent width of an absorption line and the column density of
atoms in that species is not always going to be linear. Another consequence is
something called limb darkening – that the “edges” of a star’s disk projected
on the sky are cooler and fainter than the center of the star’s disk projected on
the sky.
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